## Abstract

The description of the intersections of components of a Springer fiber is a very complex problem. Up to now only two cases have been described completely. The complete picture for the hook case has been obtained by N. Spaltenstein and J.A. Vargas, and for two-row case by F.Y.C. Fung. They have shown in particular that the intersection of a pair of components of a Springer fiber is either irreducible or empty. In both cases all the components are non-singular and the irreducibility of the intersections is strongly related to the non-singularity. As it has been shown in J. Algebra298 (2006) 1-4, a bijection between orbital varieties and components of the corresponding Springer fiber in GL_{n} extends to a bijection between the irreducible components of the intersections of orbital varieties and the irreducible components of the intersections of components of Springer fiber preserving their codimensions. Here we use this bijection to compute the intersections of the irreducible components of Springer fibers for two-column case. In this case the components are in general singular. As we show the intersection of two components is non-empty. The main result of the paper is a necessary and sufficient condition for the intersection of two components of the Springer fiber to be irreducible in two-column case. The condition is purely combinatorial. As an application of this characterization, we give first examples of pairs of components with a reducible intersection having components of different dimensions.

Original language | English |
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Pages (from-to) | 611-631 |

Number of pages | 21 |

Journal | Indagationes Mathematicae |

Volume | 19 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2008 |

### Bibliographical note

Funding Information:spondence ." This work has been varieties" and by the Marie Curie Research Training Networks E-mails:me1nikov@math.haifa.ac.iI(A.Melnikov).ngoc.pagnon@uni-duisburg-essen.de

## Keywords

- Flag manifold
- Orbital varieties
- Robinson-Schensted correspondence
- Springer fibers

## ASJC Scopus subject areas

- Mathematics (all)